A posteriori error analysis of a positivity preserving scheme for the power-law diffusion Keller–Segel model

We study a finite volume scheme approximating a parabolic-elliptic Keller–Segel system with power-law diffusion with exponent $\gamma \in [1,3]$ and periodic boundary conditions. We derive conditional a posteriori bounds for the error measured in the $L^{\infty }(0,T;H^{1}(\varOmega ))$ norm for the...

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Bibliographic Details
Published in:IMA journal of numerical analysis 2024-10
Main Authors: Giesselmann, Jan, Kolbe, Niklas
Format: Article
Language:English
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Summary:We study a finite volume scheme approximating a parabolic-elliptic Keller–Segel system with power-law diffusion with exponent $\gamma \in [1,3]$ and periodic boundary conditions. We derive conditional a posteriori bounds for the error measured in the $L^{\infty }(0,T;H^{1}(\varOmega ))$ norm for the chemoattractant and by a quasi-norm-like quantity for the density. These results are based on stability estimates and suitable conforming reconstructions of the numerical solution. We perform numerical experiments showing that our error bounds are linear in mesh width and elucidating the behavior of the error estimator under changes of $\gamma $.
ISSN:0272-4979
1464-3642
DOI:10.1093/imanum/drae073